Czech

Not applicable.

The subject knowledge on the Bachelor´s and Magister´s degree level is requested from mathematical disciplines.

Abilities leading to successful solution of some typical classes of difference equations as well as necessary theoretical knowledge and its application will be positively estimated. During half-year term students must prepare 3 essay. The final evaluation (examination) depends on assigned points (0 points is minimum, 100 points is maximum), 30 points is maximum points which can be assigned for essays. Final examination is in written and oral form and is estimated
as follows: 0- points is minimum, 70 points is maximum.

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year. Necessary condition (for final examination) are three prepared essays motivated by published papers with applications of discrete equations.

Discrete and difference equations are the mathematical base of many fields of engineering science. The purpose of this course is to develop the basic notions concerning the properties of solutions of such equations, demonstrate methods of their solution, give methods for investigation of stability of solutions, clarify their utilization in control theory and show their applications. Therefore the attention is focused on application examples and their utilization for study of stability of processes, their controllability and observability.
The ability to orientate in the basic notions and problems of discrete and difference equations. Solving problems in the areas cited in the curriculum by use of these methods. Solving given by use of modern mathematical software. Main outcomes are:

1) Ability to solve basic classes of difference equations of the first order.
2) Usage of difference equations of first order to solution of equations describing various phenomena modeled by difference equations of the first order. Transformation of differential equations to discrete equations, modeling electrical circuits by difference equations.
3) Finding of equilibria points of scalar equations, determination stability and other properties of solutions in the vicinity of equilibria.
4) Construction of cob-web diagrams for inverstigation of stability of equailibrium points.
5) Determination of stability of numerical algorithms using equilibrium points.
6) Application of basic formulae of discrete calculus.
7) Solution of homogeneous and non-homogeneous linear discrete equations of higher-order.
8) Construction of solutions of systems homogeneous and non-homogeneous difference equations of the first order.
9) Solution of linear homogeneous system of difference equations by Putzer algorithm. Finding of a particular solution.
10) Determination of stability and non-stability of nonlinear and linear discrete systems by method of a fundamental matrix and by Lyapunov method.
11) Application of Z-transform to solution of linear difference equations of higher-order and to solution of linear difference systems.
12) Detection of controllability and observability of linear discrete systems.

See the box "Literatura" 

Not applicable.

Diblík, J., Diskrétní metody v elektroinženýrství, elektronický text, Brno, 2014 (CS)
Mickens, Ronald E., Difference Equations: Theory, Applications and Advanced Topics, Third Edition, Chapman & Hall/CRC, 2016 (CS)
Oppenheim, Alan, V., Schaffer, Ronald, W., Discrete-Time Signal Processing, 3rd Edition, Pearson, 2014 (CS)

Miček, J., Jurečka, M., Moderné prostriedky implementácie metód číslicového spracovania signálov I.. EDIS, Žilina, 2013 (CS)
Sami Fadali, M., Visioli, A., Digital Control Engineering, Analysis and Design, 2nd Edition, Elsewier, AP, 2013 (CS)